Dipole field in vacuum - Cartesian vs cylindrical coordinates

[kalosu]

Hello there Meep community,

I am trying to obtain the radiation pattern of a dipole emitter using meep. My dipole is assumed to be embedded in vacuum and I am interested in using the cylindrical coordinates implementation.

I have followed the tutorial https://meep.readthedocs.io/en/latest/Python_Tutorials/Cylindrical_Coordinates/#nonaxisymmetric-dipole-sources in order to carry on with this test. I have positioned my dipole at z=0 and I have given it an offset along r, being located at $r=1.5 \Delta r$.

As far as I understand it, in a cylindrical coordinates simulation, the obtained field components, reconstructed from the different m modes, should result in equivalent cartesian coordinates field components. This means, for the plane $rz$ with $\phi=0$, $E_r = E_x$, $E_\phi = E_y$ and $E_z = E_z$.

However, I have noticed that there is a significant difference in the structure of the obtained field components with respect to the cartesian ones. In specific, I can see that whereas in my cartesian simulation (2D, defined over the x and y plane), only the $E_x$, $E_y$ and $H_z$ field components are non-zero. However, this is totally different in the cylindrical coordinates case. Below figure presents extracted far-field components from a near to far field evaluation for both cases (the cartesian and the cylindrical coordinates implementation) over a section of a circle with an equal radius of 100 and with angles spanned between 0 and 90 degrees. From these figures, it is possible to observe that more nonzero components are present throughout the different m mode solutions. Additionally, differently from the Cartesian $E_y$ field, some of these field components do not reach a value of 0 for the largest angle of 90 degrees.

Now, if I try to use the obtained cylindrical components in order to compute the far-field radiation pattern for the vacuum dipole, the obtained results highly differ from the reference analytical expression which is also included in the top right subfigure in addition to the far-field pattern obtained from the cartesian 2D simulation components. I have tested both methods presented in the https://meep.readthedocs.io/en/latest/Python_Tutorials/Cylindrical_Coordinates/#nonaxisymmetric-dipole-sources tutorial and both evaluations result in different far-field patterns with respect to the reference one.

Having said this, I would like to ask the following:

1. Do the obtained far-field components in cylindrical coordinates make sense? As previously mentioned, I would expect that at $\phi=0$, the superposition of all m modes results in the same distributions as for the cartesian case right? (Again, $E_r = E_x$, $E_y$, $E_z=E_z$). From below figure it can be seen again that this is not the case, in fact there is a significant difference with respect to the cartesian components. What could I be missing here?
   

2. Since there is a difference in the obtained far-field components, obviously the obtained Poynting flux is different and as a consequence the radiation patterns are also different. What could I do in this case? What could I be missing here??

Any feedback will be appreciated.

[kalosu]

Hello there again meep community,

I have made a few more evaluations and found some things that might be interesting. In the referred tutorial from above, it is mentioned that only one of the two modes for a given m value should be obtained since these are complex conjugate. Instead of just using one m term, I have explicitly obtained the contributions from both positive and negative terms and carried out with the same evaluation as above.

By doing this and by neglecting the contributions from the m=0 mode, the obtained far-field profiles at $\phi=0$ and the obtained far-field Poynting fluxes (again at the $\phi=0$ plane) are somehow in better agreement to the Cartesian and analytical reference results. You can see this in the following plot

Using the obtained flux components $p_r$, $p_\phi$ and $p_z$ in cylindrical coordinates and $p_x$, $p_y$ and $p_z$ in cartesian coordinates, I obtain the flux contributions to the normals of a quarter of a circle which was used for calculating the far-field components as:
$p_{cyl} = cos(\theta)p_z + sin(\theta)p_r$ in cylindrical coordinates
and
$p_{cart} = cos(\theta)p_y + sin(\theta)p_x$ in cartesian coordinates.

In this case, the obtained far-field pattern agrees in better way to the cartesian pattern and the analytical reference.
However, if I now include the contributions from the m=0 mode:

Obviously the obtained far-field pattern is highly affected by the contribution from the m=0 mode results.

Again, what could be going on here? Is there any missing scaling factor that I should consider for the m=0 mode? Its contribution is quite significant in this case. Or, for this situation, should I not consider the m=0 contribution at all?

As always, thanks for the feedback and input.